Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets

It is known that partial spreads is a class of bent partitions. In \cite{AM2022Be,MP2021Be}, two classes of bent partitions whose forms are similar to partial spreads were presented. In \cite{AKM2022Ge}, more bent partitions $\Gamma{1}, \Gamma{2}, \Gamma{1}^{\bullet}, \Gamma{2}^{\bullet}, \Theta{1}, \Theta{2}$ were presented from (pre)semifields, including the bent partitions given in \cite{AM2022Be,MP2021Be}. In this paper, we investigate the relations between bent partitions and vectorial dual-bent functions. For any prime $p$, we show that one can generate certain bent partitions (called bent partitions satisfying Condition $\mathcal{C}$) from certain vectorial dual-bent functions (called vectorial dual-bent functions satisfying Condition A). In particular, when $p$ is an odd prime, we show that bent partitions satisfying Condition $\mathcal{C}$ one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that $\Gamma{1}, \Gamma{2}, \Gamma{1}^{\bullet}, \Gamma{2}^{\bullet}, \Theta{1}, \Theta{2}$ are bent partitions. We present a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions. We show that any ternary weakly regular bent function $f: V{n}^{{(3)}\rightarrow} \mathbb{F}{3}$ ($n$ even) of $2$-form can generate a bent partition. When such $f$ is weakly regular but not regular, the generated bent partition by $f$ is not coming from a normal bent partition, which answers an open problem proposed in \cite{AM2022Be}. We give a sufficient condition on constructing partial difference sets from bent partitions, and when $p$ is an odd prime, we provide a characterization of bent partitions satisfying Condition $\mathcal{C}$ in terms of partial difference sets.